Rotation of Rigid Bodies: Angular Kinematics and Moment of Inertia

Rotation of Rigid Bodies

Introduction

Rotational motion is a fundamental aspect of physics, describing the movement of objects around a fixed axis. Examples include airplane propellers, revolving doors, ceiling fans, and Ferris wheels. In this chapter, we focus on the ideal case of perfectly rigid bodies, where deformation due to stretching or twisting is neglected.

Angular Kinematics

Angular Coordinate

• Angular position is specified by the angle θ (in radians) from a reference axis, typically the +x-axis.

• The axis of rotation is fixed and passes through the origin, perpendicular to the plane of rotation.

Units of Angles

• One radian is the angle subtended by an arc whose length equals the radius of the circle.

• One complete revolution: $360^{ ext{o}} = 2 ext{π}$ radians.

• Relationship: $\theta = \frac{s}{r}$, where $s$ is the arc length and $r$ is the radius.

Angular Velocity

• The average angular velocity is $\omega_{\text{avg},z} = \frac{\Delta \theta}{\Delta t}$, where the subscript $z$ indicates rotation about the z-axis.

• Angular displacement $\Delta \theta = \theta_2 - \theta_1$ over a time interval $\Delta t$.

• By convention, $\theta$ increases in the counterclockwise direction (positive angular velocity).

Instantaneous Angular Velocity

• The instantaneous angular velocity is $\omega_z = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t} = \frac{d\theta}{dt}$.

• Angular velocity can be positive or negative, depending on the direction of rotation.

Angular Velocity as a Vector

• Angular velocity is a vector, with direction given by the right-hand rule: curl the fingers of your right hand in the direction of rotation; your thumb points in the direction of $\vec{\omega}$.

• For rotation along the z-axis: $\omega_z > 0$ (positive z-direction), $\omega_z < 0$ (negative z-direction).

Example: Rotational Motion in Bacteria

• Escherichia coli bacteria swim by rotating their flagella at angular speeds of 200 to 1000 revolutions per minute (about 20 to 100 rad/s), with variable angular acceleration.

Angular Acceleration

• The average angular acceleration is $\alpha_{\text{avg},z} = \frac{\omega_{z,2} - \omega_{z,1}}{t_2 - t_1} = \frac{\Delta \omega_z}{\Delta t}$.

• The instantaneous angular acceleration is $\alpha_z = \frac{d\omega_z}{dt}$.

• Direction: If $\vec{\alpha}$ and $\vec{\omega}$ are in the same direction, rotation speeds up; if opposite, rotation slows down.

Rotation with Constant Angular Acceleration

The equations for rotational motion with constant angular acceleration are analogous to those for straight-line motion with constant linear acceleration:

Straight-Line Motion (Linear)	Rotational Motion (Angular)
$a_x = \text{constant}$	$\alpha_z = \text{constant}$
$v_x = v_{0x} + a_x t$	$\omega_z = \omega_{0z} + \alpha_z t$
$x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$	$\theta = \theta_0 + \omega_{0z} t + \frac{1}{2} \alpha_z t^2$
$v_x^2 = v_{0x}^2 + 2 a_x (x - x_0)$	$\omega_z^2 = \omega_{0z}^2 + 2 \alpha_z (\theta - \theta_0)$
$x - x_0 = \frac{1}{2}(v_x + v_{0x}) t$	$\theta - \theta_0 = \frac{1}{2}(\omega_z + \omega_{0z}) t$

Relating Linear and Angular Kinematics

• A point at distance $r$ from the axis of rotation has linear speed $v = r\omega$.

• Tangential acceleration: $a_{\text{tan}} = r\alpha$

• Centripetal (radial) acceleration: $a_{\text{rad}} = \frac{v^2}{r} = r\omega^2$

The Importance of Using Radians

• When relating linear and angular quantities, always use radians for angles.

• For example, $s = r\theta$ is only valid if $\theta$ is in radians.

Rotational Kinetic Energy and Moment of Inertia

Rotational Kinetic Energy

• The rotational kinetic energy of a rigid body is $K = \frac{1}{2} I \omega^2$.

• Moment of inertia $I$ is calculated as $I = \sum m_i r_i^2$ for discrete masses, or $I = \int r^2 dm$ for continuous mass distributions.

• The SI unit of $I$ is kilogram-meter squared (kg·m2).

Moment of Inertia: Physical Meaning

• Moment of inertia quantifies how mass is distributed relative to the axis of rotation.

• Mass closer to the axis → smaller $I$ → easier to start/stop rotation.

• Mass farther from the axis → larger $I$ → harder to start/stop rotation.

Example: Bird Wings

• Hummingbirds have small wings (small $I$), allowing rapid flapping (up to 70 beats/s).

• Andean condors have large wings (large $I$), resulting in slower flapping (about 1 beat/s).

Moments of Inertia for Common Bodies

Body	Axis	Moment of Inertia $I$
Slender rod	Through center	$\frac{1}{12}ML^2$
Slender rod	Through one end	$\frac{1}{3}ML^2$
Rectangular plate	Through center	$\frac{1}{12}M(a^2 + b^2)$
Thin rectangular plate	Along edge	$\frac{1}{3}Ma^2$
Hollow cylinder	Central axis	$\frac{1}{2}M(R_1^2 + R_2^2)$
Solid cylinder	Central axis	$\frac{1}{2}MR^2$
Thin-walled hollow cylinder	Central axis	$MR^2$
Solid sphere	Diameter	$\frac{2}{5}MR^2$
Thin-walled hollow sphere	Diameter	$\frac{2}{3}MR^2$

Gravitational Potential Energy of an Extended Body

• The gravitational potential energy is $U_{\text{grav}} = Mg y_{\text{cm}}$, where $y_{\text{cm}}$ is the height of the center of mass.

• Application: High jumpers arch their bodies so the center of mass passes under the bar, minimizing the required energy.

The Parallel-Axis Theorem

• Relates the moment of inertia about any axis to that about a parallel axis through the center of mass:

$I_p = I_{\text{cm}} + Md^2$

• Where $d$ is the distance between axes, $M$ is the total mass.

• Example: Calculating the moment of inertia of a baseball bat about an axis not through its center of mass.

Moment of Inertia Calculations

• For continuous mass distributions: $I = \int r^2 dV$

• Geophysicists use satellite data to measure Earth's moment of inertia, revealing that Earth's core is much denser than its outer layers.

Additional info: The notes above are structured to provide a comprehensive overview of rotational kinematics and dynamics, suitable for college-level physics students preparing for exams or seeking a concise reference.